# Independent and dependent rates formulae under UDD

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Under UDD in the multiple decrement model, we can calculate the dependent and independent rates using the below formula:

$$q_x^k=\frac{(aq)^k_x}{1-\frac{1}{2}(aq)_x^{-k}}$$

where $$q_x^k$$ represents the independent rate and $$(aq)^k_x$$ represents the dependent rate and $$(aq)_x^{-k}$$ represents the total probability of decrement (from all causes, excluding decrement k). Our textbook provides another method (under UDD), this is given below:

$$p_x^{*(j)}=(p_x^{00})^{p_x^{0j}/p_x^{0\bullet}}$$

Where the notation above has the same meaning as defined in the textbook.

Are these two formulae equivalent?

answered Nov 15, 2017 by (1,120 points)

The first formula holds if the UDD assumption is made for independent rates and there are only 2 decrements. i.e. the total probability of decrement excluding the rate of decrement for which we are calculating independent rates is comprised of only only one decrement probability.

The second formula is calculated under the assumption that dependent rates are uniformly distributed over the investigation period,

The equations are not equivalent.